Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1007 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(8,15)(12,17)(14,19)(18,20)(21,22)(23,29)(24,26)(25,32)(27,31)(28,33)(30,34)(35,36);(6,10)(7,13)(12,17)(14,19)(22,24)(27,29)(28,32)(34,35);(1,2)(3,4)(5,6,9,10)(7,11,13,16)(8,12,15,17)(14,18,19,20)(21,23,26,31)(22,27,24,29)(25,30,33,36)(28,34,32,35);(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,24,26,22)(23,29,31,27)(25,32,33,28)(30,35,36,34);(5,8)(6,12)(7,14)(9,15)(10,17)(11,18)(13,19)(16,20)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,35)(31,36)Polycyclic presentation:
2^8|a^2=fg,c^2=fh,d^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97245, #132104, #133220, … |
| maximal normal subgroup of | #97245, #132104, #133220, … |
| maximal quotient of | #553734, #553996, #554192, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29958);